Supplement to Common Knowledge
Proof of Proposition 3.4
Proposition 3.4.
In a game \(\Gamma\), common knowledge of Bayesian rationality is
satisfied if, and only if, (3.i) is common knowledge.
Proof.
Suppose first that common knowledge of Bayesian rationality is
satisfied. Since it is common knowledge that agent \(i\) knows that
agent \(k\) is Bayesian rational, it is also common knowledge that if
\(\mu_i (s_{kj}) \gt 0\), then \(s_{kj}\) must be optimal for \(k\)
given some belief over \(S_{-k},\) so (3.i) is common knowledge.
Suppose now that (3.i) is common knowledge. Then, by (3.i), agent \(i\) knows that agent \(k\) is Bayesian rational. Since (3.i) is common knowledge, all statements of the form ‘For \(i, j, \ldots ,k \in N,\) \(i\) knows that \(j\) knows that … is Bayesian rational’ follow by induction. \(\Box\)